Category Decomposition Method Based on Matched
Filter for Un-Mixing of Mixed Pixels Acquired with
Spaceborne Based Hyperspectral Radiometers
Kohei Arai 1
Graduate School of Science and Engineering Saga University
Saga City, Japan
Abstract—Category decomposition method based on matched filter for un-mixing of mixed pixels: mixels which are acquired with spaceborne based hyperspectral radiometers is proposed. Through simulation studies with simulated mixed pixels which are created with spectral reflectance data derived from USGS spectral library as well as actual airborne based hyperspectral radiometer imagery data, it is found that the proposed method works well with acceptable decomposition accuracy.
Keywords—category decomposition; hyperspectral radiometer; mixed pixel; un-mixing; matched filter;
I. INTRODUCTION
Hyperspectrometer in the visible to near infrared wavelength regions are developed and used for general purposes of earth observation missions such as Agriculture, Mineralogy, Surveillance, Physics, Chemical Imaging, Environment, in particular, for mineral resources explorations and agricultural monitoring [1]-[15]. Hyperspectormeter allows estimate atmospheric continuants by using absorption characteristics of the atmospheric continuants because spectral bandwidth of the hyperspectrometer is quit narrow like an atmospheric sounders onboard earth observation satellites [16].
Remote sensing is the practice of deriving information about the earth land and water surfaces using images acquired from an overhead perspective, using electromagnetic radiation in one or more regions of the electromagnetic spectrum, reflected or emitted from the earth surface. In particular, hyperspectral sensor (for instance, T.Lillesand et al., 1994 [17]) that covers from visible to short wave infrared wavelength region has many continuation spectrum bands (G.Vane, et al., 1993 [18], J.B.Adams, et al., 1986 [19]). Not only one single ground cover target but also two or more targets (category) are contained in the instantaneous field of view of the sensor. It is generally called as mixed pixel (Mixel) (K.Arai, 1991 [20]).
Un-mixing is the technique of presuming the category kind that constitutes the mixel, and its mixing ratio (N.Keshava, et al., 2002 [21]). There are two models for the Mixel, a linear and a nonlinear model (S.Liangrocapart et al., 1998 [22], K. Arai et al., 1992 [23]). The spectrum feature of the pixel that consists of one category is called a pure pixel (also it is called an end-member). Un-mixing is performed based on the linear or the nonlinear models (C.C.Borel et al., 1994 [24]). It is as
which a linear model disregards the interaction between end-members, and a nonlinear model considers the multiple reflection and scattering which depends on the geometric relations among the sun, a ground cover target, and a sensor (K.Arai et al., 2002 [25]). There are linear model based un-mixing methods that based on (1) a maximum likelihood method (J.J.Settle, 1996 [26], M.Matsumoto, et al., 1991 [27]), (2) a least square method with constraints (C.I.Chang, 2003 [28], K.Arai et al., 1995 [29]), (3) a spectrum feature matching (A.S.Mazer, 1988 [30]), (4) a partial space projective technique (C.Chang, et al., 1998 [31], K.Arai et al., 2002 [32]), (5) a rectangular partial space method (J.C.Harsanyi et al., 1994 [33]), etc. The least square method with a constraint presumes a mixing ratio vector based on an end-member's spectrum feature vector by the generalized inverse matrix or the least-squares method which makes convex combination conditions a constraint. The spectrum feature matching searches and selects two or more spectrum features out of a plenty of spectrum features in a spectral database. It is the spectral feature matching method in consideration of those mixing ratios, and the spectrum feature of the Mixel in concern.
consideration an end-member's spectrum feature and its variation. The statistic model about a component is considered and presumption of a statistic model and the ratio of each component are presumed by unsupervised learning. Therefore, it is the method of using the independent component analysis with constraints. Moreover, since the orthogonal subspace method becomes ideally independent in the spectrum feature after projection, it is also equivalent to ICA. From the above reason, this paper shall examine the un-mixing based on the orthogonal subspace method. The subspace method with learning process is already proposed as the image classification technique (Oja Erkki, 1983 [36]). The basic idea for that is the following. If the axis of coordinates of the subspace in the orthogonal subspace method is rotated, then classification accuracy will be improved to find an appropriate angle for a high classification performance, and then the spectrum feature of a pixel is mapped and classified into the orthogonal subspace of this optimal axis-of-coordinates angle. The following section describes the proposed category decomposition with matched filter method followed by some experiments. Then conclusion is described together with some discussions.
II. PROPOSED METHOD
A. Conventional Un-Mixing Method
Hyper-spectral data represented by vector Y can be expressed as follows:
mZ z z z z z z z m m m y y y Y mn m n n m n . . . . . . . . . . . . . . . . . . . . . 1 2 2 1 1 1 2 1 1 2 1 2 1 (1)
where m denotes mixing ratio vector of each ground cover target no.1 to m, while Z denotes the spectral characteristics of the ground cover target such that if the inverse matrix of Z is existing then the mixing ratio vector can be estimated as follows,
1
YZ
m (2)
It is, however, not always true that the inverse matrix exists. In order to solve this problem, regularization techniques with constraints, a prior information, etc., have been proposed. One of those is the generalized inverse matrix, or so called
“Moore-Penrose” inverse matrix that is derived from Singular
Value Decomposition (SVD). If Y is expressed with SVD as follows:
Yv u
X t
(3) where u and v are orthogonal vectors, then the Moore-Penrose generalized inverse matrix Y+ is expressed by the following equation:
Yu v
Y t (4)
Therefore, if u and v can be calculated then the generalized inverse matrix can also be calculated. This method is referred to the conventional SVD based method hereafter.
However, the number of spectral bands, n is more than 200 for hyperspectral imaging sensors, so that a time-consuming matrix calculation is required. In order to overcome this situation, the subspace method (SSM) is introduced.
If an n-dimensional observation data Y is mapped into an n-dimensional feature space of X, in general, then the square of the norm of Y is mapped to that of X as follows:
X p X X p p X X p i t i i t
i
2
(5) where pi denotes the orthogonal mapping matrix so that piX
can be mapped from X to the subspace. Thus the mixing ratio of ground cover target, j, can be estimated with the following equation:
n j j j t j j i t X p X m X p X 1 (6)If the dominant dimensions are selected from the subspace, then SSM can also reduce the dimensionality of the feature space. It may be said that the first three dimensions would cover more than 90% of the whole information, following the subspace method of conversion of orthogonal mapping. This can reduce time-consuming computations for the matrix calculations. By using the probability density function of the mapped observation vector, un-mixing can be done without any time-consuming calculation. The subspace method is closely related to the well known PCA (Principal Component Analysis) analysis which allows convert feature space coordinate into the principal coordinate system using rotation of coordinate system. In this case the original feature space, X, is mapped into the subspace, u. On the other hand, the proposed subspace method adjust the rotation angle to concentrate the information content, the rotation angle is adjusted through an iterative learning process. The most appropriate rotation angle is determined by category by category. Generally, PCA determines an appropriate rotation angle in the sense of average means. On the other hand, the proposed method takes separability between all the combinations of categories
By using the definition of length which is expressed with the equation (5),
) ... ( ) ( )
( 1 1 1 1 1 1 1 i m 1 m
t t X p m X p m X p X p X
p
.. ) ... ( ) ( )
( 1 m 1 m m m
t m m m t m
mX p X p X m p X m p X
p (7)
then, 1 , 1 1 1 1 , 1 1 1 1 1
1X cos m || pX ||cos ... mm ||pXm ||cos m
p
. m m m m m m m m m m m
mX m p X m p X
p cos || 1||cos ,1... || ||cos ,
(8)
where Xi denotes feature vector of category i while X
denotes unknown vector which has to be estimated its mixing ratio. Feature vector Xi of category i can be mapped onto
subspace piXi. The angle between its coordinate, piXi and
orthogonal transformation of pkXk is αi,i,k. Thus mixing ratio,
B. Matched Filter
Hyper-spectral radiometer has more than one hundred spectral channels. Therefore, it requires, in general, a huge computer resources for category decomposition, or un-mixing which is composed with huge element size of matrix calculus. On the other hand, there is a matched filter which allows extract specific object which has a specific spectral feature from mixed spectral characteristics of object. In this section, background theory of matched filter is introduced. If the specific spectral feature is known a prior basis, then the matched filter can be used for category decomposition. In other word, if there is intensive ground cover material, mixing ratio of the material in the mixed pixel in concern can be estimated with the proposed matched filter.
In the time domain, output signal y[n] is expressed as equation (1),
(1) where x and h denotes input signal and impulse response function of the system. Input signal consists of signal and noise as shown in equation (2).
(2) where v denotes noise and noise power is expressed as equation (3).
(3)
where H denotes complex conjugate. Then output can be represented as equation (4)
(4) The signal-to-noise ratio SNR is expressed as equation (5)
(5) Signal component can be expressed as equation (6).
(6) Nose component, on the other hand, is expressed with equation (7).
(7) Thus signal-to-noise ratio becomes the following equation,
(8) Assuming the following equation,
(9) Then cost function can be expressed with the following equation using Lagrange multiplier, λ,
(10)
From equation (10), the following equations are derived, (11) (12) This is well known as a generalized eigen value problem.
(13)
Since is of unit rank, it has only one nonzero eigen value. Therefore, it can be shown that this eigen value equals
(14) which is yielding the following optimal matched filter
(15)
C. Un-Mixing Method
The mixed pixel: Mixel in concern is assumed to be composed with several ground cover materials. Figure 1 shows mathematical model of the Mixel. Namely, the Mixel is composed with more than two spectral characteristics are combine together depending on their mixing ratios. Using spectral characteristic of ground cover material in concern, it is possible to extract same spectral characteristic from the combined spectral characteristics using matched filter.
Two types of noises, colored and white noise are added to the pure pixel of spectral characteristics.
Fig. 1. Spectral characteristic of Mixel model
If the proposed matched filter is applied to the Mixel data with a assumed spectral feature of pure pixel in concern, then mixing ratio can be estimated as shown in Figure 2.
(b)Time domain
Fig. 2. Mixing ratio estimation from the Mixels by using the specific spectral feature of the assumed ground cover materials in concern
Matched filter can be applied in frequency and time domains. Using Fourier transformation, the proposed un-mixing method based on matched filter can be expressed in both time and frequency domains as shown in Figure 3.
Fig. 3. Proposed un-mixing method based on matched filter which is expressed in both time and frequency domains
III. EXPERIMENTS AND SIMULATIONS
A. Simulation Method
From the Unites State of America Geological Survey: USGS web site so called “Spectral Library”, spectral characteristics (surface reflectance) can be retrieved for huge number of ground cover materials. In the library, two ground cover materials which show a good correlation between both spectral characteristics are chosen. Also two ground cover materials which show a poor correlation between both are selected. Moreover, ground cover materials which show a middle level of correlation are used. These are shown in Figure 4. 437 spectral channels ranged from 500 nm to 2500 nm of wavelength region of spectral characteristics are used.
(a)Poor Correlation (R=0.0009)
(b)Middle Level of Correlation (R=0.5)
(a)Best Correlation (R=0.99)
Fig. 4. Examples of the selected two ground cover materials of spectral characteristics which show a good a middle level and a poor correlation between both spectral characteristics
Using these spectral characteristics, un-mixing by means of the proposed matched filter based un-mixing method is attempted with changing mixing ratio as well as additive noises.
Root Mean Square Error: RMSE between designated and estimated mixing ratios is evaluated. The RMSE is evaluated for both the conventional SVD based method and the proposed method.
B. Simulation Results
Fig. 5. Evaluated RMSE for both the conventional and the proposed methods as function of cross correlation between spectral characteristics extracted from the USGS spectral library
As shown in Figure 5, it is clear that RMSE for the proposed method is always smaller than that of the conventional method.
RMSE is also depending on the number of material of the Mixels used for simulation studies. Figure 6 shows RMSE as function of the number of ground cover materials of which the Mixels used are composed.
RMSE for the conventional un-mixing method varied greatly in comparison to that of the proposed method and is greater than that of the proposed method. It is concluded that there is a poor relation between RMSE and the number of ground cover materials of which the Mixels used for simulation. RMSE depends on the complexity of spectral characteristics of the ground cover materials and also depends on the correlation among the materials. RMSE is not function of the number of materials. Therefore, RMSE of the conventional method varied a lot comparing to the proposed method. The proposed method extracts the spectral characteristics from the mixed spectral characteristics so that RMSE is not varied too much.
Fig. 6. Relation between the evaluated RMSE and the number of ground cover materials of mixels used for the simulation
Required computational time for un-mixing based on the proposed method, on the other hand, increases in accordance with the number of ground cover materials with which the Mixels used for simulation are composed as shown in Figure 7.
Fig. 7. Required computational time for un-mixing, on the other hand, increases in accordance with the number of ground cover materials with which the Mixels used for simulation are composed
Figure 8 shows the evaluated RMSE with the parameters of mixing ratio, signal to noise ratio and cross correlation for both the conventional and the proposed un-mixing methods. Two materials are mixed together when the Mixels used for simulation is created. There are three levels of correlation coefficients between two materials. Those are 0.0009, 0.5, and 0.99.
C. Experimental Results with Actual Airborne Based Hyper
Spectrometer of AVIRIS
Actual hyper-spectral sensor data (AVIRIS) onboard aircraft is used for validation of the proposed method. Figure 9 shows the AVIRIS imagery data of Ivanpah playa in California, USA. The site is covered with silica Cray mostly.
From the image, silica Cray of pixel is extracted at the
x=54, y=479, while asphalt pixel is also extracted from the
pixel location at x=81, y=405, respectively. Meanwhile, four test pixels are extracted from the pixels in the Ivanpah playa. These pixels are situated on the Root # 15 of road so that the pixels are essentially covered with asphalt. Then the mixing ratio of silica Cray and Asphalt are estimated with the proposed Matched Filter: MF based method and the conventional Singular Value Decomposition: SVD based method. Estimated mixing ratios are shown in Table 1.
(a)SVD based method
(b)Proposed matched filter based method
Fig. 8. Evaluated RMSE with the parameters of mixing ratio, signal to noise ratio and cross correlation for both the conventional and the proposed un-mixing methods
TABLE I. ESTIMATED MIXING RATIO OF SILICA CRAY AND ASPHALT ARE ESTIMATED WITH THE PROPOSED MATCHED FILTER:MF BASED METHOD
AND THE CONVENTIONAL SINGULAR VALUE DECOMPOSITION:SVD BASED METHOD
Pixel Silica Cray Asphalt
Location SVD MF SVD MF
Data #1 (96,402) 0.32 0.21 0.54 0.76
Data #2 (108,402) 0.36 0.22 0.40 0.73
Data #3 (128,402) 0.33 0.21 0.51 0.75
Data #4 (295,405) 0.36 0.21 0.47 0.73
Fig. 9. AVIRIS imagery data of Ivanpah playa in California, USA.
IV. CONCLUSION
Category decomposition method based on matched filter for un-mixing of mixed pixels: Mixels which are acquired with spaceborne based hyper-spectral radiometers is proposed. Through simulation studies with simulated mixed pixels which are created with spectral reflectance data derived from USGS spectral library as well as actual airborne based hyper-spectral radiometer imagery data, it is found that the proposed method works well with acceptable decomposition accuracy
ACKNOWLEDGMENT
The author would like to thank Mr. Tadashi Yokota for his effort to conduct experiments.
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AUTHORS PROFILE
